Page 40 - Chemical engineering design

23
INTRODUCTION TO DESIGN
With this order, the equations can be solved sequentially, with no need for the simul-
taneous solution of any of the equations. The fortuitous selection of v 3 and v 4 as design
variables has given an efficient order of solution of the equations.
If for a set of equations an order of solution exists such that there is no need for the
simultaneous solution of any of the equations, the system is said to be “acyclic”, no
recycle of information.
If another pair of variables had been selected, for instance v 5 and v 7 , an acyclic order
of solution for the set of equations would not necessarily have been obtained.
For many design calculations it will not be possible to select the design variables so as
to eliminate the recycle of information and obviate the need for iterative solution of the
design relationships.
For example, the set of equations given below will be cyclic for all choices of the two
possible design variables.

f 1 x 1 ,x 2 D 0
f 2 x 1 ,x 3 ,x 4 D 0
f 3 x 2 ,x 3 ,x 4 ,x 5 ,x 6 D 0

f 4 x 4 ,x 5 ,x 6 D 0
N d D 6 4 D 2
The biparte graph for this example, with x 3 and x 5 selected as the design variables
(inputs), is shown in Figure 1.13.

x 3 x 5

f 1 f 2 f 3 f 4

x 1 x 2 x 4 x 6

Figure 1.13.

One strategy for the solution of this cyclic set of equations would be to guess (assign
avalue to) x 6 . The equations could then be solved sequentially, as shown in Figure 1.14,
to produce a calculated value for x 6 , which could be compared with the assumed value
and the procedure repeated until a satisfactory convergence of the assumed and calculated
value had been obtained. Assigning a value to x 6 is equivalent to “tearing” the recycle
loop at x 6 (Figure 1.15). Iterative methods for the solution of equations are discussed by
Henley and Rosen (1969).
When a design problem cannot be reduced to an acyclic form by judicious selection of
the design variables, the design variables should be chosen so as to reduce the recycle of